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        <title>Ideal (Ring Theory)</title>

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                <h1 id="title" titleSize="">
                    Ideal (Ring Theory)
                </h1>
            
            <h1 id="motivation--definition">Motivation &amp; Definition</h1>
<p>Fix a <a href=https://zhaoshenzhai.github.io/mathwiki/ring.md class="internalLink references" title="Ring" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/ring.md&#34;, {&#34;Date&#34;:&#34;2024-06-06T15:26:58-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-06-06T15:26:58-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-06-06T15:26:58-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Ring&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/ring&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onmouseleave="clearPreviewSide({&#34;Date&#34;:&#34;2024-06-06T15:26:58-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-06-06T15:26:58-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-06-06T15:26:58-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Ring&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/ring&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/ring.md&#34;, {&#34;Date&#34;:&#34;2024-06-06T15:26:58-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-06-06T15:26:58-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-06-06T15:26:58-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Ring&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/ring&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});">ring</a> $R$. Its underlying <a href=https://zhaoshenzhai.github.io/mathwiki/abelian_group.md class="internalLink references" title="Abelian Group" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/abelian_group.md&#34;, {&#34;Date&#34;:&#34;2024-05-22T18:53:29-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-05-22T18:53:29-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-05-22T18:53:29-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Abelian Group&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/abelian_group&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onmouseleave="clearPreviewSide({&#34;Date&#34;:&#34;2024-05-22T18:53:29-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-05-22T18:53:29-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-05-22T18:53:29-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Abelian Group&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/abelian_group&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/abelian_group.md&#34;, {&#34;Date&#34;:&#34;2024-05-22T18:53:29-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-05-22T18:53:29-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-05-22T18:53:29-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Abelian Group&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/abelian_group&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});">abelian group</a> $G$ admits a notion of <a href=https://zhaoshenzhai.github.io/mathwiki/congruence_relation.md class="internalLink references ghostLink" title="congruences" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/congruence_relation.md&#34;, &#34;nopPage&#34;);" onmouseleave="clearPreviewSide(&#34;nopPage&#34;);" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/congruence_relation.md&#34;, &#34;nopPage&#34;);">congruences</a>, which are <a href=https://zhaoshenzhai.github.io/mathwiki/equivalence_relation.md class="internalLink references" title="Equivalence Relation" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/equivalence_relation.md&#34;, {&#34;Date&#34;:&#34;2024-06-18T20:07:21-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-06-18T20:07:21-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-06-18T20:07:21-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Equivalence Relation&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/equivalence_relation&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onmouseleave="clearPreviewSide({&#34;Date&#34;:&#34;2024-06-18T20:07:21-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-06-18T20:07:21-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-06-18T20:07:21-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Equivalence Relation&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/equivalence_relation&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/equivalence_relation.md&#34;, {&#34;Date&#34;:&#34;2024-06-18T20:07:21-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-06-18T20:07:21-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-06-18T20:07:21-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Equivalence Relation&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/equivalence_relation&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});">equivalence relations</a> $\sim$ on $G$ that respect the group structure, and those correspond exactly to<a href=https://zhaoshenzhai.github.io/mathwiki/normal_subgroup.md/#normal_iff_congruence class="internalLink justifications dag" title="Normal Subgroup" mathLink="" secID="normal_iff_congruence" secDisplay="Normal $\Leftrightarrow$ Congruence" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/normal_subgroup.md/#normal_iff_congruence&#34;, {&#34;Date&#34;:&#34;2024-06-10T17:20:55-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-06-10T17:20:55-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-06-10T17:20:55-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Normal Subgroup&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/normal_subgroup&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onmouseleave="clearPreviewSide({&#34;Date&#34;:&#34;2024-06-10T17:20:55-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-06-10T17:20:55-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-06-10T17:20:55-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Normal Subgroup&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/normal_subgroup&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/normal_subgroup.md/#normal_iff_congruence&#34;, {&#34;Date&#34;:&#34;2024-06-10T17:20:55-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-06-10T17:20:55-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-06-10T17:20:55-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Normal Subgroup&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/normal_subgroup&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});">$^\dagger$</a> <a href=https://zhaoshenzhai.github.io/mathwiki/normal_subgroup.md class="internalLink references" title="Normal Subgroup" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/normal_subgroup.md&#34;, {&#34;Date&#34;:&#34;2024-06-10T17:20:55-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-06-10T17:20:55-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-06-10T17:20:55-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Normal Subgroup&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/normal_subgroup&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onmouseleave="clearPreviewSide({&#34;Date&#34;:&#34;2024-06-10T17:20:55-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-06-10T17:20:55-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-06-10T17:20:55-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Normal Subgroup&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/normal_subgroup&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/normal_subgroup.md&#34;, {&#34;Date&#34;:&#34;2024-06-10T17:20:55-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-06-10T17:20:55-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-06-10T17:20:55-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Normal Subgroup&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/normal_subgroup&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});">normal</a> <a href=https://zhaoshenzhai.github.io/mathwiki/subgroup.md class="internalLink references" title="Subgroup" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/subgroup.md&#34;, {&#34;Date&#34;:&#34;2024-06-06T19:40:13-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-06-06T19:40:13-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-06-06T19:40:13-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Subgroup&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/subgroup&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onmouseleave="clearPreviewSide({&#34;Date&#34;:&#34;2024-06-06T19:40:13-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-06-06T19:40:13-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-06-06T19:40:13-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Subgroup&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/subgroup&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/subgroup.md&#34;, {&#34;Date&#34;:&#34;2024-06-06T19:40:13-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-06-06T19:40:13-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-06-06T19:40:13-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Subgroup&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/subgroup&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});">subgroups</a> of $G$. Since $G$ is abelian, they correspond to all subgroups.</p>
<br>
<p>  However, the presence of the multiplication operation motivates us to consider congruences on $G$ that also respect multiplication, in that $r_1\sim r_2$ and $s_1\sim s_2$ implies $r_1s_1\sim r_2s_2$. It turns out that those congruences correspond directly to special subgroups of $G$ (<em>not</em> <a href=https://zhaoshenzhai.github.io/mathwiki/subring.md class="internalLink references ghostLink" title="subrings" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/subring.md&#34;, &#34;nopPage&#34;);" onmouseleave="clearPreviewSide(&#34;nopPage&#34;);" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/subring.md&#34;, &#34;nopPage&#34;);">subrings</a> of $R$), namely, the <em>ideals</em>:</p>
<div class="space"></div>
<p><strong>Remark.</strong> Just like how $G$ can be viewed as a <a href=https://zhaoshenzhai.github.io/mathwiki/groupoid.md class="internalLink references ghostLink" title="groupoid" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/groupoid.md&#34;, &#34;nopPage&#34;);" onmouseleave="clearPreviewSide(&#34;nopPage&#34;);" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/groupoid.md&#34;, &#34;nopPage&#34;);">groupoid</a> with a single object, $R$ can be viewed as a <a href=https://zhaoshenzhai.github.io/mathwiki/preadditive_category.md class="internalLink references ghostLink" title="preadditive category" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/preadditive_category.md&#34;, &#34;nopPage&#34;);" onmouseleave="clearPreviewSide(&#34;nopPage&#34;);" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/preadditive_category.md&#34;, &#34;nopPage&#34;);">preadditive category</a> with a single object. Then, the ‘special congruences’ discussed above are the actual congruences in this category. <span style="color:red"><strong>TODO</strong> (verify this).</span></p>
<div class="env envDef" id=""><img class="icon noSelect listenDark" src="https://zhaoshenzhai.github.io/mathwiki/css/fa/definition.svg"><b class="envTitle">Definition. </b><p>Let $R$ be a ring. A <em>left ideal</em> in $R$ is a subgroup $(\mf{a},+)\substructeq R$ such that $r\mf{a}\subseteq\mf{a}$ for all $r\in R$.</p>
<br>
<p>  Similarly for <em>right</em> ideals and <em>(two-sided) ideals</em>. If $\mf{a}\substructeq R$ is an ideal, we write $\mf{a}\nsubgrpeq R$.</p>
</div>

<p>Essentially the same proof that normal subgroups correspond to congruences for groups works here:</p>
<blockquote>
<div class="collapsibleContainer" id=""><i class="proofHeader collapsibleHeaderButton collapsibleHeader noSelect hide">Proof.</i><span class="collapsibleHintText noSelect"><i> Click to expand...</i></span>

        <span class="collapsibleContent"><p>Given a congruence $\sim$, the subset $\mf{a}\coloneqq\l\{r\in R\st n\sim0\r\}$ is a subgroup of $R$. For all $r\in R$, we have $rn\sim r0=0$, so $r\mf{a}\subseteq\mf{a}$. Similarly, we have $\mf{a}r\subseteq\mf{a}$, so $\mf{a}$ is a two-sided ideal.</p>
<br>
<p>  Conversely, given a two-sided ideal $\mf{a}\nsubgrpeq R$, the relation $\sim$ on $R$ defined by $r_1\sim r_2$ iff $r_1-r_2\in\mf{a}$ is an equivalence relation respecting addition. It respects multiplication too, since if $r_1\sim r_2$ and $s_1\sim s_2$, then $r_1-r_2\in\mf{a}$ and $s_1-s_2\in\mf{a}$, and hence
$$\begin{equation}
r_1s_1-r_2s_2=r_1s_1-r_1s_2+r_1s_2-r_2s_2=r_1(s_1-s_2)+(r_1-r_2)s_2\in\mf{a}.\qedin
\end{equation}$$</p>
</span></div>

</blockquote>
<p>Ideals $\mf{a}\nsubgrpeq R$ turn out to be exactly the objects one can <a href=https://zhaoshenzhai.github.io/mathwiki/quotient_ring.md class="internalLink references" title="Quotient Ring" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/quotient_ring.md&#34;, {&#34;Date&#34;:&#34;2024-08-29T17:01:19-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-08-29T17:01:19-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-08-29T17:01:19-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Quotient Ring&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/quotient_ring&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onmouseleave="clearPreviewSide({&#34;Date&#34;:&#34;2024-08-29T17:01:19-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-08-29T17:01:19-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-08-29T17:01:19-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Quotient Ring&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/quotient_ring&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/quotient_ring.md&#34;, {&#34;Date&#34;:&#34;2024-08-29T17:01:19-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-08-29T17:01:19-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-08-29T17:01:19-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Quotient Ring&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/quotient_ring&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});">quotient</a> $R$ with so that $\pi:R\onto R/\mf{a}$ is a ring homomorphism. In this case, we have $\ker\pi=\mf{a}$, and conversely, the kernel of any ring homomorphism $f:R\to S$ is an ideal. This gives us a three-fold equivalence between congruences of $R$, ideals in $R$, and kernels of homomorphisms $f:R\to S$.</p>
<br>
<p>  If $R$ is commutative, then special types of ideals reflect special properties of quotients of $R$. Most notably, an ideal $\mf{p}\nsubgrpeq R$ is <em><a href=https://zhaoshenzhai.github.io/mathwiki/prime_and_maximal_ideals.md class="internalLink types" title="Prime and Maximal Ideals" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/prime_and_maximal_ideals.md&#34;, {&#34;Date&#34;:&#34;2024-08-30T16:11:08-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-08-30T16:11:08-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-08-30T16:11:08-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Prime and Maximal Ideals&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/prime_and_maximal_ideals&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onmouseleave="clearPreviewSide({&#34;Date&#34;:&#34;2024-08-30T16:11:08-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-08-30T16:11:08-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-08-30T16:11:08-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Prime and Maximal Ideals&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/prime_and_maximal_ideals&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/prime_and_maximal_ideals.md&#34;, {&#34;Date&#34;:&#34;2024-08-30T16:11:08-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-08-30T16:11:08-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-08-30T16:11:08-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Prime and Maximal Ideals&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/prime_and_maximal_ideals&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});">prime</a></em> iff $R/\mf{q}$ is an <a href=https://zhaoshenzhai.github.io/mathwiki/integral_domain.md class="internalLink references ghostLink" title="integral domain" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/integral_domain.md&#34;, &#34;nopPage&#34;);" onmouseleave="clearPreviewSide(&#34;nopPage&#34;);" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/integral_domain.md&#34;, &#34;nopPage&#34;);">integral domain</a>, and an ideal $\mf{m}\nsubgrpeq R$ is <em><a href=https://zhaoshenzhai.github.io/mathwiki/prime_and_maximal_ideals.md class="internalLink types" title="Prime and Maximal Ideals" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/prime_and_maximal_ideals.md&#34;, {&#34;Date&#34;:&#34;2024-08-30T16:11:08-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-08-30T16:11:08-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-08-30T16:11:08-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Prime and Maximal Ideals&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/prime_and_maximal_ideals&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onmouseleave="clearPreviewSide({&#34;Date&#34;:&#34;2024-08-30T16:11:08-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-08-30T16:11:08-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-08-30T16:11:08-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Prime and Maximal Ideals&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/prime_and_maximal_ideals&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/prime_and_maximal_ideals.md&#34;, {&#34;Date&#34;:&#34;2024-08-30T16:11:08-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-08-30T16:11:08-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-08-30T16:11:08-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Prime and Maximal Ideals&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/prime_and_maximal_ideals&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});">maximal</a></em> iff $R/\mf{m}$ is a <a href=https://zhaoshenzhai.github.io/mathwiki/field.md class="internalLink references" title="Field" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/field.md&#34;, 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onmouseleave="clearPreviewSide({&#34;Date&#34;:&#34;2024-08-31T19:11:12-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-08-31T19:11:12-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-08-31T19:11:12-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Field&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/field&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/field.md&#34;, {&#34;Date&#34;:&#34;2024-08-31T19:11:12-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-08-31T19:11:12-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-08-31T19:11:12-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Field&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/field&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});">field</a>.</p>
<h2 id="generation-of-ideals">Generation of Ideals</h2>
<p>Just like how any subset $A\subseteq G$ of a group $G$ <a href=https://zhaoshenzhai.github.io/mathwiki/subgroup.md/#subgroup_generation class="internalLink references" title="Subgroup" mathLink="" secID="subgroup_generation" secDisplay="Subgroup Generation" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/subgroup.md/#subgroup_generation&#34;, {&#34;Date&#34;:&#34;2024-06-06T19:40:13-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-06-06T19:40:13-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-06-06T19:40:13-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Subgroup&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/subgroup&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onmouseleave="clearPreviewSide({&#34;Date&#34;:&#34;2024-06-06T19:40:13-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-06-06T19:40:13-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-06-06T19:40:13-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Subgroup&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/subgroup&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/subgroup.md/#subgroup_generation&#34;, {&#34;Date&#34;:&#34;2024-06-06T19:40:13-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-06-06T19:40:13-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-06-06T19:40:13-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Subgroup&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/subgroup&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});">generates</a> a subgroup $\gen{A}\substructeq G$, subsets $A\subseteq R$ of a ring $R$ generates an ideal in the same way. We first need to note that an intersection $\bigcap_\alpha\mf{a}_\alpha$ of (left/right-)ideals is still an (left/right-)ideal.</p>
<ul>
<li><span style="color:gray">(Explicitly).</span> The most direct construction is by letting $\gen{A}$ be the set of all finite sums $\sum_{i&lt;n}r_ia_is_i$ for some $a_i\in A$ and $r_i,s_i\in R$, which makes $\gen{A}\nsubgrpeq R$. To generate a left ideal $RA$ (resp. right ideal $AR$), we only include the sums $\sum_{i&lt;n}r_ia_i$ (resp. $\sum_{i&lt;n}a_is_i$).</li>
<li><span style="color:gray">(Top-down).</span> We can let $\gen{A}$ be the intersection of all (left/right) ideals of $R$ containing $A$.</li>
<li><span style="color:gray">(Bottom-up).</span> We can construct $\gen{A}$ starting with $A$ and inductively closing it under addition and (left/right-)multiplication by $R$.</li>
</ul>
<p>An ideal $\mf{a}\nsubgrpeq R$ is said to be <em>principal</em> if $\mf{a}$ is generated by single element. More generally, $\mf{a}$ is <em>finitely-generated</em> if $\mf{a}=\gen{a_1,\dots,a_n}$.</p>
<h2 id="arithmetic-of-ideals">Arithmetic of Ideals</h2>
<p>Other than the intersection, there are several other operations of ideals. The two most important ones are their <em>sum</em> and their <em>product</em>.</p>
<ol>
<li>For an arbitrary family $\mf{\alpha}_\alpha\nsubgrpeq R$ of ideals, their <em>sum</em> is the ideal $\sum_\alpha\mf{a}_\alpha$ consisting of all finite sums $\sum_\alpha r_\alpha$ where $r_\alpha\in\mf{a}_\alpha$ vanishes for cofinitely-many $\alpha$. Equivalently, it is the ideal generated by $\bigcup_\alpha\mf{a}_\alpha$.</li>
<li>Their <em>product</em> of ideals $\mf{a},\mf{b}\nsubgrpeq R$ is the ideal $\mf{a}\mf{b}$ generated by the set of all products $ab$ for $a\in\mf{a}$ and $b\in\mf{b}$. Similarly for the product of <em>finitely-many</em> ideals (but <em>not</em> for infinite products). It is clear that $\mf{a}\mf{b}\subseteq\mf{a}\cap\mf{b}$, but the converse is not necessarily true.</li>
</ol>


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                June 6, 2024 | Zhaoshen Zhai

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